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Asymptotic lp spaces and bounded distortions

arXiv:math/9209214v1 [math.FA] 15 Sep 1992Asymptotic lp spaces and bounded distortionsVitali D. MilmanNicole Tomczak-JaegermannAbstractThe new class of Banach spaces, so-called asymptotic lp spaces,is introduced and it is shown that every Banach space with boundeddistortions contains a subspace from this class. The proof is basedon an investigation of certain functions, called enveloping functions,which are intimately connected with stabilization properties of thenorm.0IntroductionDuring the last year several problems of infinite-dimensional Banach spacetheory, which remained open for decades, have been finally solved.

Some newconstructions of Banach spaces have been made which, on one hand, showedlimitations of the theory, but on the other hand, also showed how excitingan infinite-dimensional geometry can be. Let us mention few of them:(i) a space without unconditional basic sequence (Gowers–Maurey),(ii) a space not isomorphic to any of its hyperplanes (Gowers),(iii) a space such that every bounded operator being a Fredholm operator(Gowers–Maurey).The problems which were answered by these examples are of a linear-topological nature.

Although a thorough study of this kind of propertiesflourished back in the 60s, methods developed that time and later were notsufficient to succesfully atack these problems. The solutions given last yearare by-products of a study in a different direction: the infinite-dimensional1

geometry of convex bodies, that is, the geometry of the unit sphere of aBanach space.In this introduction we would like to explain the geometry which led tothe breakthrough described above; and in the main body of the paper wewould like to add some information in this geometric direction.0.1In fact, the topic of studies which led to the recent development takesits roots, in a large part, in the local theory of Banach spaces, in other words,in the asymptotic theory of finite-dimensional normed spaces. Consider thefollowing question:Let f(·) be a uniformly continuous real valued function on the unit sphereS = S(X) = {x ∈X | ∥x∥= 1} of an infinite-dimensional Banach space X.Does the oscillation of f decrease to zero on some sequence En of infinite-dimensional subspaces of X?To state it in a more precise way we need some notation.

For a fixedfunction f as above, and for an arbitrary subspace E ⊂X, let IE(f) =[a(E), b(E)], where a(E) = inf{f(x) | x ∈S ∩E} and b(E) = sup{f(x) | x ∈S ∩E}. Then letO(f) = OX(f) = inf{b(E) −a(E) | E ⊂X, dim E = ∞}.The question then becomes: is O(f) = 0?If the answer is “yes” thenthere exists a real number s such that for every ε > 0 there is a subspace Ewith dim E = ∞such that |f(x) −s| < ε for all x ∈S ∩E.The collection of all the numbers s is called the spectrum of f and denotedby γ∞(f) or γ∞(f, X) (see [M.69]).And so, we are asking whether the spectrum γ∞(f) is non-empty for alluniformly continuous functions f on the sphere of an arbitrary Banach spaceX, or of some Banach space X?Intuition says that the answer is obviously negative, at least for X = l2,say, because there is no reason for it to be positive.

Uniform continuity is alocal geometric condition with no connection to a linear structure of a space,and the existence of s ∈γ∞(f) is a global linear property. One never studieswhat seems to be obvious and the question was not an exception to this rule.2

Note that James [J.64] showed that, in the above terminology, γ∞(f, l1)and γ∞(f, c0) are non-empty for f being an equivalent norm on these spaces.This result did not contradict the intuition, because the norms in l1 and c0are in a sense extremal, and the proofs deeply depended on this fact. So, atthe time, it did not even raise a similar question for, say, l2.0.2However, it was observed in 1967 ([M.67], cf.

also [M.69], [M.71a])that a slightly different finite-dimensional spectrum γ(f) is always non-empty.We say that s ∈γ(f) wheneverfor every ε > 0 and for every n there exists an n-dimensional subspaceEn ⊂X such that |f(x) −s| < ε for all x ∈S ∩En.We have the following fact valid for every infinite-dimensional Banach spaceX.FactFor every uniformly continuous real function f on the unit sphere S,γ(f) ̸= ∅.As we explained above, this somewhat contradicted intuitions of thattime. Just to support these intuitions, let us recall the Grinblatt’s paper[G.76] where an example was presented of a bounded continuous, but notuniformly continuous, function f on the sphere S in the Hilbert space, whichhas the oscillation at least 1 on every 2-dimensional central section of S.So the fact above indeed fundamentally rests on an interplay of uniformcontinuity of a function and non-compactness of the sphere.Thus, since the finite-dimensional spectrum γ(f) involves subspaces ofarbitrarily high dimensions and it is always non-empty, it eventually becamenatural to expect that the (infinite-dimensional) spectrum γ∞(f) is also non-empty.0.3Let us now consider the case when the function f =·is anothernorm on X, continuous with respect to the original norm.We have thefollowing two mutually exclusive possibilities.

(a) Spectrum: For every norm f we have γ∞(f) ̸= ∅. This would meanthat either on some infinite-dimensional subspace f is arbitrarily small,if 0 ∈γ∞(f), or, if 0 ̸= s ∈γ∞(f), then f is “almost” an isometry onsome infinite-dimensional subspace.3

(b) Distortion: There is a norm f such that γ∞(f) = ∅. This means thatthe norm f has an oscillation with respect to the original norm non-decreasing to zero on any infinite-dimensional subspace.In view of the Fact above, the existence of a norm satisfying condition(b) would be clearly connected with some very essential infinite-dimensionaleffects.If a uniformly continuous function f satisfies (b) then, obviously, thereexists an interval I = [β, δ], with β < δ, such that(i) for every ε > 0 there exists a subspace Y = Yε, dim Y = ∞, such thatIY (f) ⊂[β −ε, δ + ε];(ii) For every E ⊂Y , dim E = ∞one has IE(f) ⊃(β, δ).

(Here Y is asubspace from (i) corresponding to ε = 1, say. )The collection of all such intervals I is called the tilda-spectrum of f anddenoted by ˜γ(f).Of course, the case δ = β reduces the interval to one point, β ∈γ∞(f),which we also consider as a part of ˜γ(f).Therefore we have (see [M.69])FactFor every uniformly continuous real function f on the unit sphere S,˜γ(f) ̸= ∅.0.4Note that if f is a norm on X as in 0.3, and if I = [0, δ] ∈˜γ(f), thennecessarily δ = 0 (see [M.69]).In the case β > 0 we introduce a level of distortion of an interval I ∈˜γ(f)by d(I) = δ/β, and a level of distortion of an equivalent norm f byd(f) = sup{d(I) | I ∈˜γ(f)}.We have a similar alternative as in 0.3.

(a’) Either for any equivalent norm f on X one has d(f) = 1,(b’) or there exists an equivalent norm f on X such that d(f) > 1.4

In terms of the spectrum, condition (a’) means that for any equivalentnorm f on X and any infinite-dimensional subspace Z of X, the spectrumγ∞(f|Z) of the restriction of f to Z, in non-empty. Similarly, condition (b’)means that X contains a distortable infinite-dimensional subspace: thereexists an infinite-dimensional subspace Z of X and an equivalent norm f onX such that γ∞(f|Z) = ∅, that is, f is a distortion on Z.It was proved by Milman in 1969 thatTheoremLet X be a Banach space.Assume that d(f) = 1 for everyequivalent norm f on X.

Then either for some 1 ≤p < ∞, X contains a(1 + ε)-isomorphic copy of lp (for every ε > 0), or X contains a (1 + ε)-isomorphic copy of c0 (for every ε > 0). (The result was stated in [M.69], Section 3.3, with the complete proof in[M.71b].

)And so, alternative (a’) would imply an exciting structural theory forBanach spaces. However, in 1974, Tsirelson [Ts.74] constructed a space Twhich does not contain an isomorphic copy of any lp (1 ≤p < ∞) or ofc0.

This means that the space T satisfies the alternative (b’): T contains adistortable infinite-dimensional subspace Z. (In fact, it can be shown by adirect argument that T itself is also distortable.

)An interesting feature of Tsirelson’s example is that the norm is not givenby an explicit formula but it is defined by an equation. This was the firstconstruction of such a type, and essentially, with only minor modifications,the only one.

In the dual form, which has been put forward by Figiel andJohnson [F-J.74], the norm is defined, for a finite sequence of real numbersx ∈IR (IN), by∥x∥T = max(∥x∥c0, 12 supnXi=1∥Eix∥T),(0.1)where the inside supremum is taken over all succesive intervals {Ei} of pos-itive integers such that n < min E1 ≤max E1 < min E2 ≤. .

. < max En−1

Tsirelson’s space T is then a completion of IR (IN) under thenorm ∥· ∥T. Most of important properties of the space T and related spacescan be found in [C-S.89] and references therein.5

0.5Let us return to a distortion situation when I = [β, δ] ∈˜γ(f) withβ < δ and let us give its geometric interpretation.Let ε < (δ −β)/2 and let Y = Yε be a corresponding subspace. Definetwo setsA = {x ∈S ∩Y | f(x) < β + ε}andB = {x ∈S ∩Y | f(x) > δ −ε}.For every infinite-dimensional subspace E ⊂Y we have A ∩E ̸= ∅andB∩E ̸= ∅.

A set satisfying such a property is called an asymptotic set (in Y ).So in our situation, A and B are two asymptotic sets with positive distanceapart, dist (A, B) > 0. The fact of the existence of such a pair (A, B) is thusa consequence of distortion.

Conversely, this fact also implies some distortionproperty. The Urysohn function for sets A and B is a uniformly continuousfunction with an empty spectrum γ∞(f); to construct an equivalent normwithout spectrum some additional convexity assumptions are required.0.6Given a distortion situation it is natural to ask a quantitative ques-tion, how large can d(f) be.

Note that Theorem 0.4 ensures only the existenceof distortion but provides no quantitative information on d(f).It is to Rosenthal’s credit that in 1988 he asked the first named author,Odell and several others, how to find a direct formula for a distortion onTsirelson’s space, and how large such a distortion can be. Odell (unpub-lished) in 1989/90 constructed two asymptotic sets in T. He also showedthat the spaces Tλ, obtained by replacing 1/2 in the definition (0.1) by 1/λ,have distortions dλ of order 1/λ, hence dλ →∞as λ →∞.

So, for everyreal number d, there is a space with a level of distortion at least d.Let us mention that an approach to distortions using the theory of Krivine–Maurey types was presented in [H-O-R-S.91]. In particular this paper con-tains another proof of Theorem 0.4.The next step was done by Schlumprecht [S.91], who changed 1/2 to1/ ln n, which also allowed to start E1 at any place (not necessarily far out).This had an important effect on the geometry of the space: the unit vectorbasis becames subsymmetric and, as Schlumprecht showed, any distortionlevel is attained by some equivalent norm.

Schlumprecht’s space S also hasthe property of an infinite distortion:there exists a sequence of asymptotic sets {Ai} on the sphere of Ssuch that dist (Ai, conv (Sj̸=i Aj)) ≥1.6

In fact, S satisfies still stronger condition that there also exists a sequence ofsets {A∗i } on the sphere of the dual space S∗such that the system {Ai, A∗i }is “nearly biorthogonal”.This was the starting point for Gowers’ and Maurey’s construction.Finally, this year, Odell and Schlumprecht [O-S.92] proved that for every1 < p < ∞, lp has an arbitrarily large (and even infinite) distortion, this wayfinishing offthe problem which originated from [M.69], [M.71b]. Again, theydid not construct asymptotic sets far apart in, say, l2, but transformed themin an ingeneous non-linear way from Tsirelson’s space, or, on more advancedlevel, from Schlumprecht’s space.

Combining this outstanding result withTheorem 0.4 we see thatTheoremAny Banach space X which does not hereditarily contain copiesof l1 and c0, contains a distortable subspace, i.e., there exists an equivalentnorm f on X such that d(f) > 1.Moreover, Odell and Schlumprecht proved that on the sphere S(l1) thereis a Lipschitz function f (not a norm) with an empty spectrum, γ∞(f, l1) = ∅.It was shown earlier by Gowers [G.91] that γ∞(f, c0) ̸= ∅, for any uniformlycontinuous function on S(c0).0.7Let us go back to the quantitative question: does any Banach spaceX not containing hereditarily copies of l1 and c0, have arbitrarily large dis-tortions?This is not yet clear. To study this problem, we consider in this paperspaces with bounded distortions.These are spaces X such that for someconstant D we have d(f) ≤D, for every infinite-dimensional subspace Z ofX and every equivalent norm f on Z.

What kind of simple “basic structuralblocks” (i.e., subspaces) can such a space X contain? To explain our resultlet us define the class of asymptotic lp spaces.

(The rather standard notationconcerning successive blocks of a basis and related concepts will be explainedat the beginning of the next section. )DefinitionA Banach space X with a normalized basis {xi} is said to beasymptotic lp space, for some 1 ≤p < ∞(resp.

asymptotic c0 space) ifthere exists a constant C such that for every n there exists N = N(n) suchthat any normalized successive blocks N < z1 < z2 < . .

. < zn of {xi} are7

C-equivalent to the unit vector basis in lnp (resp. in ln∞).

By λp(X) we denotethe infimum of all constants C as above.Note that Tsirelson’s space T is an asymptotic l1 space which does notcontain a subspace isomorphic to l1.For spaces with bounded distortions, letd(X) = sup d(f),(0.2)where the supremum is taken over all equivalent norms f on X.Recall a standard and easy observation that if Z ⊂X is an infinite-dimensional subspace and f is an equivalent norm on Z then there exists anequivalent norm ˜f on X such that ˜f|Z = f. This immediately implies thatd(Z) ≤d(X).TheoremLet X be a Banach space with bounded distortions and let d(X)

Moreover, λp(Y ) depends on D only.We learned recently that B. Maurey [Ma.92] also proved this theorem andused it to show that every space of type p > 1 with an unconditional basishas arbitrarily large distortions.In contrast with the result for d(X) = 1 (Theorem 0.4), the theorem aboverecognizes, as “basic structural blocks”, a class of Banach spaces rather thana concrete space, as it was suggested by a “naive” intuition of the 60s. (Itis well-known that varying λ in the definition of Tsirelson’s spaces Tλ weget a sequence of non-isomorphic asymptotic l1 spaces, and the so-called p-convexified Tsirelson’s spaces show that the same phenomenon holds for anyfixed 1 ≤p < ∞or c0.

)Another important point is a difference with the local theory of Banachspaces.The definition of asymptotic lp spaces is “almost” local, in thatit involves finite-dimensional subspaces and parameters not depending onthe dimension. However, this isomorphic definition does not imply a (1 + ε)-isometric version, in the standard spirit of the local theory.

Indeed, we wouldcall a Banach space an almost isometric asymptotic lp space if λp(X) = 1. Itis well-known to specialists that every almost isometric asymptotic lp spacecontains, for every ε > 0, a subspace (1 + ε)-isomorphic to lp (see 6.4 for ashort argument).Our method involves geometry of infinite-dimensional sphere and a suit-able geometric language will be introduced in the next section.8

1Preliminaries1.1Since in this paper we are concerned with the existence of nice infinite-dimensional subspaces inside Banach spaces from a certain class, we may andwill assume, unless stated otherwise, that Banach spaces discussed here havea monotone basis. In such a situation we will use the standard notions of thedual basis, equivalent bases, block bases, block subspaces, basic sequences,etc.

They can be found e.g., in [L-T.77]. Let us only mention that we will saythat two basic sequences {xi} and {ei} are C-equivalent, for some constantC, if for any (finite) sequence of scalars {ai} we haveC−1∥Xiaixi∥≤∥Xiaiei∥≤C∥Xiaixi∥.We will consider only vectors with finite support.

For vectors x, y ∈X andsubspaces E, E1, E2, we will freely use the notation n < x to denote thatn < min supp (x); then x < y if max supp (x) < min supp (y); then x < E ifx < y for every y ∈E; and E1 < E2 if x < y for every x ∈E1 and y ∈E2.For a Banach space X by BX and S(X) we denote the unit ball and theunit sphere in X, respectively; for a subspace E ⊂X we set BE = BX ∩Eand S(E) = S(X) ∩E.1.2Asymptotic sets were defined in 0.5 where their basic connection todistortions was indicated. To get a better understanding of their geometricproperties let us make some easy general observations, valid for arbitraryBanach spaces (which may have no basis).FactLet (Z, ∥·∥) be a Banach space and let·be a seminorm on Z suchthatz≤∥z∥, for all z ∈Z.

Assume that there exists an asymptotic setA ⊂S(Z, ∥· ∥) such that ∥· ∥and·are equivalent on A. Then thereexists a subspace E of Z of finite-codimension such that·is a norm onE equivalent to ∥· ∥.ProofClearly,·is a norm on the subspace W spanned by the set A,and since A is asymptotic then codim W < ∞.

A standard well-known fact(cf. e.g., [K.66], [L-T.77]) implies that if·and ∥·∥were not equivalent onany subspace E of W of finite-codimension then for every ε > 0 there wouldbe an infinite-dimensional subspace F of W such thatz≤ε∥z∥, for allz ∈F.

But for ε sufficiently small this is impossible, since F intersects A. ✷9

Remark Let Z be a Banach space with bounded distortions, d(Z) < D, andlet A ⊂S(Z) be an asymptotic set symmetric about the origin. Then thereexists an infinite-dimensional subspace F of Z such that (1/D)(BZ ∩F) ⊂conv A.Indeed, let W be the finite-codimensional subspace spanned by A. Ap-plying the fact above to the norm | · | on W whose unit ball is conv A ∩W,we get that ∥· ∥and | · | are equivalent on a certain subspace E of W offinite codimension.

Thus there exists an infinite-dimensional subspace F ofE such that∥z∥≤|z| ≤D∥z∥forz ∈F. (1.1)This implies the required inclusion of the corresponding unit balls.1.3To make the arguments more compact, we introduce several shortnotations for certain families of subspaces of a given Banach space Z (witha basis).

Typically, Z will be a block subspace of the fixed Banach space X.By B ∞(Z) we denote the family of all infinite-dimensional block sub-spaces E ⊂Z; next, B t(Z) denotes the family of all (block) subspacesE ∈B ∞(Z) of finite-codimension, i.e., dim Z/E < ∞; finally, if Y ∈B ∞(Z)and z ∈S(Z) (with finite support), then B t(Y, z) denotes the family of allsubspaces F ∈B ∞(Y ) such that z < w for all w ∈F.1.4Let us recall the geometric notions of asymptotic averages and moduli,which play a major role in our approach. These notions were introduced andstudied by Milman in 1967–70.

A survey on this subject can be found in[M.71b], cf. also more recent paper [M-P.89].The moduli are defined relatively to a fixed family B of subspaces of aspace X, which satisfies the filtration conditionFor every E1, E2 ∈B there exists E3 ∈B such that E3 ⊂E1 ∩E2.Typically, the family B will be B t, which have been defined in 1.3, althoughwe will make an exception from this rule in Section 2.For a continuous bounded function h : S(X) →IR and an infinite-dimensional subspace E ⊂X define lower and upper moduli β- and δ-,respectively, byβ[h, B , E]=βx[h, B (E)] =supF ∈B (E)infx∈S(F )h(x),10

δ[h, B , E]=δx[h, B (E)] =infF ∈B (E)supx∈S(F )h(x). (1.2)For a continuous bounded function f : S(X) × S(X) →IR and E ∈B ∞(X)we setββ[f, B , E] = βx [βy[f(x, ·), B (E, x)], B , E]δδ[f, B , E] = δx [δy[f(x, ·), B (E, x)], B , E] .

(1.3)Observe that if E ⊂F thenβ[h, B , F] ≤β[h, B , E] ≤δ[h, B , E] ≤δ[h, B , F],hence alsoββ[f, B , F] ≤ββ[f, B , E] ≤δδ[f, B , E] ≤δδ[f, B , F],for all functions h and f as above.2Non-distortable spacesTo develop better geometric intuitions and to illustrate the use of the β- andδ- averages we start with the isometric case and we will sketch the proof ofMilman’s theorem on non-distortable spaces, Theorem 0.4.2.1In the isometric situation discussed here there is no real advantage inpassing to a subspace with a basis, in fact, this would confuse a geometricpicture rather than clarify it. Therefore we present an argument which makesno reference to the existence of a basis and thus it works for an arbitraryBanach space.

The averages we will consider here will be taken with respectto the family B = B 0(E) of all finite-codimensional subspaces of a givenspace E. This family clearly satisfies the filtration condition.We will consider the collection of functionsfε(x, y) = ∥x + εy∥−1forx, y ∈S(X). (2.1)The averages of functions fε will be called the β- and δ-moduli, as theyreflect a geometric behaviour of the sphere in a Banach space.11

For a subspace E ⊂X and x ∈S(X), the notation B (E, x) used in(1.3) simply means B 0(E). Also, B ∞(E) denotes the family of all infinite-dimensional subspaces of E.We will consider the local modulus βy[fε(x, ·), B 0(E, x)] denoting it byβ(ε, x, E) and the global modulus ββ[fε, B 0, E], denoting it by ββ(ε, E).Similarly, the local modulus δy[fε(x, ·), B 0(E, x)] will be denoted by δ(ε, x, E)and the global modulus δδ[fε, B 0, E], by δδ(ε, E).To illustrate the expected behaviour of the moduli, let us observe thatfor X = lp, 1 ≤p < ∞we haveβ(ε, x, lp) = δ(ε, x, lp) = (1 + εp)1/p −1,for all x ∈S(lp) and all ε > 0.

This function has the order εp/p as ε →0.Computation of the moduli for some other spaces can be found in [M.71b].2.2LemmaLet X be a Banach space such that d(X) = 1. There exists aninfinite-dimensional subspace F of X such thatββ(ε, F) = δδ(ε, F)forε > 0.

(2.2)Proof It is not difficult to see that if d(f) = 1 for every equivalent norm fon Z then OZ(g) = 0 for every Z ∈B ∞(X) and every uniformly continuousconvex function g : Z →IR . We will show this at the end of the proof.Observe that for each ε > 0 and x ∈S(X), the function fε(x, ·) is convex,therefore O(fε(x, ·)) = 0.

Stabilizing over y with a given θ > 0 we get asubspace ˜E ∈B ∞(X) such that0 ≤supy∈S( ˜E)fε(x, y) −infy∈S( ˜E)fε(x, y) < θforx ∈S( ˜E). (2.3)Now we take a dense set {xi} in the unit sphere S(X) and a sequenceof θi ↓0, and we let, for every i = 1, 2, .

. ., ε vary over a finite θi-net Niin [θi, 1/θi]; this way we can construct a sequence E1 ⊃.

. .

⊃Ei ⊃. .

. ofinfinite-dimensional subspaces such that on S(Ei) the inequality analogousto (2.3) holds for xj, with j = 1, .

. .

, i, and θi and all ε ∈Ni, (i = 1, 2, . .

. ).Picking ei ∈S(Ei) for i = 1, 2, .

. .

and setting E = span [ei], we get, by this12

diagonal procedure, E ∈B ∞(X) such that for all x ∈S(E) and ε > 0 wehaveβ(ε, x, E) = δ(ε, x, E).Now, with a fixed ε > 0, the function δ(ε, ·, E) is again convex. Stabilizingover x, with a fixed ε > 0, and then passing to a diagonal in a similar way asbefore, we get an infinite-dimensional subspace F of E on which (2.2) holds.It remains to show that OZ(g) = 0 for every Z ∈B ∞(X) and everyuniformly continuous convex function g : Z →IR .

To this end, fix Z ∈B ∞(X) and pick [β, δ] ∈˜γ(g), and, for an arbitrary (fixed) ε > 0, let Y ∈B ∞(Z) be a corresponding stabilizing subspace, satisfying conditions (i) and(ii) from 0.3.Assume first that g(x) = g(−x) for g ∈S(X). Consider the symmetricasymptotic sets A and B defined in 0.5.

By Remark in 1.2 (with D = 1+ε) weget a subspace E ∈B ∞(Y ) such that B∩E ⊂BE = BX ∩E ⊂(1+ε)conv A.Thus, by convexity and uniform continuity of g, we haveδ −ε −θ ≤infz∈B∩E g(z) ≤supw∈A g(w) + θ ≤β + ε + θ,where θ = θ(ε) →0 as ε →0. Letting ε →0 we get β = δ, hence OZ(g) = 0.If g is arbitrary, set h(x) = (1/2)(g(x) + g(−x)).

Since OY (h) = 0, find asubspace Y1 ∈B ∞(Y ) such that |h(x) −s| < ε for all x ∈S(Y1). If ε is smallenough then |s −(β + δ)/2| < 2ε, hence β + ε < s < δ −ε.

(In fact, we willuse the later inequality only.) Indeed, otherwise g(x) and g(−x) would notcompensate each other.

Formally, pick w, v ∈S(Y1) such that g(w) = β + εand g(v) = δ −ε and observe thats −(β + δ)/2 ≤(s −h(w)) + ((g(w) + g(−w))/2 −(β + δ)/2) ≤2ε;and similarly, using h(v), it is easy to establish the lower estimate by −2ε.Consider the set A = {y ∈S(Y1) | |g(y) −s| ≤ε}, which is asymptoticin Y1. Observe that for y ∈A we have|g(−y) −s| ≤|g(y) + g(−y) −2s| + |g(y) −s| ≤3ε,so that A is “almost” symmetric.By Remark in 1.2 we get a subspaceE ∈B ∞(Y1) such that BE ⊂(1 + ε)conv (A∪−A).

Thus for every z ∈S(E)we have,g(z) ≤supw∈A∪−A g(w) + θ ≤s + 3ε + θ,13

where θ = θ(ε) →0 as ε →0. This, combined with the stabilization propertyof h, yields that OZ(g) is arbitrarily small, hence equal to 0, as required.

✷RemarkIn the non-distortion situation of the lemma it is not difficult toshow that X contains a subspace with a basis {ui} such that for every n andfor all blocks n < w < v we have max (∥w +v∥, ∥w −v∥) ≤(1+2−n)∥w +v∥.A standard argument shows that the tails of {ui} are unconditional with theconstants as close to 1 as we wish. We could then consider the moduli relatedto the family B = B t and note that the argument from 2.3 applies for thesemoduli as well.

Moreover, in this situation it would be clearly sufficient todiscuss the observation opening the proof of the lemma only for symmetricfunctions.2.3Sketch of the proof of Theorem 0.4 We will show that the equal-ity ββ(ε, F) = δδ(ε, F) yields that the space F contains (1 + ε)-isomorphiccopies of lp or c0.Using definitions of ββ(ε, F) and δδ(ε, F) it is possible, given η > 0,to construct a basic sequence {yi} in F such that for any finite sequence{ai} ∈IR (IN) of real numbers the norm ∥Pi aiyi∥admits an upper estimateby (1+η)Φ({δδ(|ai|, F)}) and a lower estimate by (1+η)−1Φ({ββ(|ai|, F)}),where Φ({·}) is a real function defined on the space of all finite sequences ofreal numbers (see [M.71b], Theorem 4.5). Moreover, the same estimates aresatisfied for every vector of the formPi aiui, where {ui} is a block basis of{yi}.

(This argument is similar to a well-known construction sketched in 6.4. )By Lemma 2.2, this implies that all block bases of {yi} are (1+η)-equivalent.By Zippin’s theorem, the basis {yi} is (1 + η)α-equivalent to the unit vectorbasis in lp, for some 1 ≤p < ∞or in c0 (here α > 0 is a numerical constant).✷3Tilda-spectrum in general3.1Let X be a Banach space with a basis {xi}.

Let f(z1, z2, . .

. , zl) bea uniformly continuous real function defined on sequences of l normalizedblock vectors z1 < z2 < .

. .

< zl. First let us describe a rough intuition of an14

interval [β, δ] of tilda-spectrum of f on a subspace ˜Y ∈B ∞(X), leaving theprecise definition for later parts of this section.Let ˜I be the closure of the interval of values of f on ˜Y . By restricting thedomain of the variable zl to any subspace of ˜Y , with other variables fixed,we do not increase ˜I.

Therefore, for any z1 < z2 < . .

. < zl−1 fixed, let Y1 bea subspace of ˜Y such that passing with zl to Y1 corresponds to the “maximaldecrease” of ˜I.

Let I(1) be the closure of the interval of values of this restrictedf. Continue the procedure of restricting zl−1, with z1 < z2 < .

. .

< zl−2fixed. The closed interval I(l) = [β, δ] obtained after the l-th step is calledan interval of the tilda-spectrum of f in ˜Y .3.2Let ˜Y ∈B ∞(X).

The precise definition of the tilda-spectrum of f on˜Y involves the notions of the β- and δ- averages, introduced in 1.4. Theseaverages will be applied to functions of the form h(z1, .

. .

, zk), considered asfunctions of zk with z1 < . .

. < zk−1 fixed, and with respect to the familyB t(Y, zk−1) of all finite-codimensional block subspaces of Y with the supportafter zk−1 (Y ∈B ∞(X) is a subspace).

To make the formulas more com-pact, we will indicate the variable zk and the subspace Y in the subscripts,leaving zk−1 to be understood from the context. Thus we will write, e.g.βzk,Y (h(z1, .

. .

, zk−1, zk)) for β[h(z1, . .

. , zk−1, ·), B t(Y, zk−1)], and so on.We say that an interval [β, δ] is in the tilda spectrum of f on ˜Y if there isa subspace Y ∈B ∞( ˜Y ) such that the following two conditions are satisfied:(i) β = βz1,Y (βz2,Y (.

. .

(βzl,Y (f(z1, z2, . .

. , zl))) .

. .

))andδ = δz1,Y (δz2,Y (. .

. (δzl,Y (f(z1, z2, .

. .

, zl))) . .

. ));(ii) for all H1, H2, .

. .

, Hl ∈B ∞(Y ), each of the averages βzi,Y and δzi,Y in(i) can be replaced by βzi,Hi and δzi,Hi, respectively; that is, we haveβ = βz1,H1 (βz2,H2(. .

. (βzl,Hl(f(z1, z2, .

. .

, zl))) . .

. ))andδ = δz1,H1 (δz2,H2(.

. .

(δzl,Hl(f(z1, z2, . .

. , zl))) .

. .

)).A subspace Y for which the above conditions hold is called a spectrumsubspace corresponding to [β, δ]. Then any further subspace Y ′ of Y is aspectrum subspace as well.15

3.3To prove the existence of the tilda-spectrum defined in 3.2, it is con-venient to introduce modified averages βst and δst. The definition requiresseveral steps.3.3.1Let h(z1, .

. .

, zk) be a uniformly continuous function and let E ∈B ∞(X). Fix normalized blocks z1 < .

. .

< zk−1 and setγ(E) = inf(δzk,G(h(z1, . .

. , zk−1, zk)) −βzk,G(h(z1, .

. .

, zk−1, zk))),where the infimum is taken over all subspaces G ∈B ∞(E). Pick εi ↓0 andconstruct a sequence G0 = E ⊃G1 ⊃.

. ., such that Gi ∈B ∞(Gi−1) andδzk,Gi(h(z1, .

. .

, zk−1, zk)) −βzk,Gi(h(z1, . .

. , zk−1, zk)) ≤γ(Gi−1) + εi, (3.1)for i = 1, 2, .

. .. Setβstzk,E(h)=βstzk,E(h; z1, .

. .

, zk−1))= limi→∞βzk,Gi(h(z1, . .

. , zk−1, zk)),δstzk,E(h)=δstzk,E(h; z1, .

. .

, zk−1))= limi→∞δzk,Gi(h(z1, . .

. , zk−1, zk)).

(3.2)Let G = span [ui] be a diagonal subspace for {Gi}, that is, ui ∈Gi fori = 1, 2, . .

.. It is easy to check from (3.1) that for every subspace H ∈B ∞(G)we haveβstzk,E(h) = βzk,H(h)andδstzk,E(h) = δzk,H(h).

(3.3)3.3.2Now let {(z(i)1 , . .

. , z(i)k−1)} be a dense countable subset of (k−1)-tuplesof normalized blocks z1 < .

. .

< zk−1. Let G(1) = G be the subspace con-structed at the end of 3.3.1 for (z(1)1 , .

. .

, z(1)k−1). Starting from G(1), constructG(2) ∈B ∞(G(1)) for the tuple (z(2)1 , .

. .

, z(2)k−1). Proceeding by induction andusing (3.3) we get a sequence of subspaces E ⊃G(1) ⊃G(2) ⊃.

. .

such thatfor every i = 1, 2, . .

. and all H, H′ ∈B ∞(G(i)) we haveβzk,H(h(z(i)1 , .

. .

, z(i)k−1, zk))=βstzk,E(h; z(i)1 , . .

. , z(i)k−1)= βzk,H′(h(z(i)1 , .

. .

, z(i)k−1, zk)),δzk,H(h(z(i)1 , . .

. , z(i)k−1, zk))=δstzk,E(h; z(i)1 , .

. .

, z(i)k−1)= δzk,H′(h(z(i)1 , . .

. , z(i)k−1, zk)).16

Taking once more a diagonal subspace we get F = span [vi], with vi ∈G(i)for i = 1, 2, . .

. such that for all (k −1)-tuples (z1, .

. .

, zk−1) and for allsubspaces H ∈B ∞(F) we haveβzk,H(h(z1, . .

. , zk−1, zk))=βstzk,E(h; z1, .

. .

, zk−1)δzk,H(h(z1, . .

. , zk−1, zk))=δstzk,E(h; z1, .

. .

, zk−1). (3.4)3.3.3Coming back to the definition 3.2 of the tilda-spectrum, fix a functionf = f(z1, .

. .

, zl) and a subspace ˜Y ∈B ∞(X).The existence of a tilda-spectrum interval [β, δ] will be proved by provid-ing explicit formulae for β and δ in terms of the stabilized averages βst andδst. This is done by the backward induction.Letb1(z1, .

. .

, zl−1)=βstzl, ˜Y (f; z1, . .

. , zl−1)d1(z1, .

. .

, zl−1)=δstzl, ˜Y (f; z1, . .

. , zl−1),and let F1 ∈B ∞( ˜Y ) be the subspace constructed at the end of 3.3.2 forwhich (3.4) is satisfied.Repeat the procedure inside F1 by settingb2(z1, .

. .

, zl−2)=βstzl−1,F1(b1; z1, . .

. , zl−2)d2(z1, .

. .

, zl−2)=δstzl−1,F1(d1; z1, . .

. , zl−2),and let F2 ∈B ∞(F1) be the corresponding subspace.Proceed by an obvious induction to get functions bi and di for i = 1, .

. .

, land subspaces ˜Y ⊃F1 ⊃. .

. ⊃Fl.

Setβ = bl=βstz1,Fl−1(bl−1)=βstz1,Fl−1βstz2,Fl−2. .

.βstzl, ˜Y (f; z1, . .

. , zl−1).

. .,δ = dl=δstz1,Fl−1(dl−1)=δstz1,Fl−1δstz2,Fl−2.

. .δstzl, ˜Y (f; z1, .

. .

, zl−1). .

..(3.5)It is easy to see, using (3.4), that with these definitions of β and δ, theinterval [β, δ] satisfies conditions (i) and (ii) of 3.2 for the subspace Y = Fl.17

3.4Using the definition of the β- and δ- averages it is easy to see thatcondition (i) of the definition of the tilda-spectrum in 3.2 is equivalent to thefollowing:(i’) ∀θ > 0 ∃Y1 ∈B t(Y ) ∀z1 ∈S(Y1) ∃Y2 ∈B t(Y1, z1) ∀z2 ∈S(Y2)∃Y3 ∈B t(Y2, z2) ∀z3 ∈S(Y3) . .

. ∃Yl ∈B t(Yl−1, zl−1) ∀zl ∈S(Yl)f(z1, z2, .

. .

, zl) ∈[β −θ, δ + θ].We will show below that this condition implies in fact a stronger property,that is, the existence of a stabilizing subspace for f. Given an interval [β, δ]satisfying condition (i) on a subspace Y we can construct, for any θ > 0, asubspace G ∈B ∞(Y ) such that for all normalized blocks z1 < z2 < . .

. < zlin G we havef(z1, z2, .

. .

, zl) ∈[β −θ, δ + θ]. (3.6)Fix θ′ > 0 and η > 0 to be defined later.

Let E1 = Y1 ∈B t(Y ) be thesubspace satisfying condition (i’) for θ′. Pick an arbitrary vector u1 ∈S(E1),and let E2 = Y2 ∈B t(Y1, u1) be the subspace from condition (i’) (again forθ′).

Pick an arbitrary vector u2 ∈S(E2).In the next step we would like to find a subspace E3 ∈B t(E2, u2) whichwould satisfy condition (i’) in several ways: it could be taken as Y3, forvectors z1 = u1 and z2 = u2, and it could be taken as Y2, for an arbitraryvector z1 running over some finite η-net N (in the original norm) on thesphere S(span [u1, u2]). Since subspaces appearing in (i’) are always of finitecodimension, it is clear that a required subspace E3 exists.

Then pick anarbitrary u3 ∈S(E3).Continuing in an obvious manner we construct a subspace G = span [ui]such that f(z1, z2, . .

. , zl) ∈[β−θ′, δ+θ′] for all z1 < .

. .

< zl with zis runningover all finite η-nets on the spheres S(span [z1, . .

. , zk]) (with k = 2, 3, .

. .

).Choosing suitable η > 0 and θ′ > 0 depending on θ, we complete the proofof (3.6).Remark Given a sequence θn ↓0 we can repeat the above construction forevery n and then pass to a diagonal subspace. We then obtain a subspaceZ ∈B ∞(Y ) with a (block) basis {vi} such that for every n and for arbitrarynormalized blocks n < z1 < .

. .

< zl of {vi} we havef(z1, . .

. , zl) ∈[β −θn, δ + θn].18

3.5The role of condition (ii) of the definition of the tilda-spectrum is toensure the existence of large sets of vectors on which the value of the functionf is close to extremal. In fact, these sets turn out to be asymptotic in somestabilizing subspace for f.Let us start by observing that condition (ii) from 3.2 is equivalent to thefollowing:(ii’) for all η > 0 and all subspaces H1, H2, .

. .

, Hl ∈B ∞(Y ) we have:∃w1 ∈S(H1) ∃w2 ∈S(H2), w2 > w1 . .

.∃wl ∈S(Hl), wl > wl−1f(w1, w2, . .

. , wl) ≤β + ηand∃v1 ∈S(H1) ∃v2 ∈S(H2), v2 > v1 .

. .∃vl ∈S(Hl), vl > vl−1f(v1, v2, .

. .

, vl) ≥δ −η.Let [β, δ] be in the tilda-spectrum of f and let Z be the correspondingstabilizing subspace constructed in Remark 3.4. Condition (ii’) leads to thenatural definition of sets asymptotic in Z.With a fixed η > 0 define A1 ⊂S(Z) byA1={w1 ∈S(Z) | ∀H2, .

. .

, Hl ∈B ∞(Z)∃w2 ∈S(H2), w2 > w1 ∃w3 ∈S(H3), w3 > w2 . .

.∃wl ∈S(Hl), wl > wl−1 f(w1, w2, . .

. , wl) ≤β + η}.By (ii’), the set A1 has a non-empty intersection with every subspaceH1 ∈B ∞(Z), hence A1 is asymptotic in Z.By induction, let 1 ≤k < l, and assume that for any fixed w1 < .

. .

. , k −1, the set Ak = Ak(w1, .

. .

, wk−1) hasbeen defined by the formulaAk={wk ∈S(Z) | ∀Hk+1, . .

. , Hl ∈B ∞(Z)∃wk+1 ∈S(Hk+1), wk+1 > wk ∃wk+2 ∈S(Hk+2), wk+2 > wk+1 .

. .∃wl ∈S(Hl), wl > wl−1 f(w1, w2, .

. .

, wl) ≤β + η}. (3.7)Moreover, assume that Ak is asymptotic in Z.

Then for any fixed w1 < . .

.

. .

, k, define Ak+1 by the formula analogous to(3.7). It clearly follows from the form of Ak that Ak+1 is asymptotic in Z.Similarily, we can define sets Uk ⊂S(Z) for j = k, .

. .

, l, which are alsoasymptotic in Z, and such that if v1 < . .

. < vl and vi ∈Ui for i = 1, .

. .

, lthen f(v1, . .

. , vl) ≥δ −η.19

3.6The notion of tilda-spectrum has the following unconditionality prop-erty.For a given function f and a finite sequence ε = (ε1, ε2, . .

.) withε1 = ±1, ε2 = ±1, .

. ., define the function fε byfε(z1, z2, .

. .

, zl) = f(ε1z1, ε2z2, . .

. , εlzl),for normalized blocks z1 < .

. .

, zl. Then if [β, δ] is in the tilda-spectrum off and Y is a corresponding spectrum subspace, then for all the fεs, [β, δ] isagain a spectrum interval with the same spectrum subspace Y .

Moreover,the stabilizing subspace Z ⊂Y of 3.4 is also preserved for all the fεs. Notehowever, that the asymptotic sets Ai and Ui described in 3.5 are not thesame.3.7The final important step in our discussion of tilda-spectrum is anobservation that a construction of stabilizing subspaces in 3.4 can be done“almost” simultaneously for any countable family of uniformly continuousreal functions, fk = fk(z1, .

. .

, zlk).For a subspace ˜Y ∈B ∞(X) and asequence θk ↓0, there exists Z ∈B ∞( ˜Y ) such that for every n = 1, 2, . .

., ifLn = maxk≤n lk, then for arbitrary normalized blocks n < z1 < . .

. < zLn inZ we havefk(z1, .

. .

, zlk) ∈[βk −θn, δk + θn]for1 ≤k ≤n.Here [βk, δk] is an interval in the tilda-spectrum of fk in ˜Y . Moreover, all thesets A(k)iand U(k)ifor i = 1, 2, .

. ., constructed in 3.5 for the function fk, areasymptotic in Z.This follows from 3.4 and 3.5 by the standard diagonal procedure.

Thedetails are left for the reader.4Spaces with bounded distortionsWe now pass to the main theorem on spaces with bounded distortions, The-orem 0.7.4.1By passing to an infinite-dimensional subspace of X and consideringa suitable renorming of X we may assume, without loss of generality, that X20

has a monotone basis. The proof of the theorem relies on stabilization prop-erties of a family of real functions which we introduce now and fix throughoutthe rest of the argument.

This family is indexed by the set Q(IN)+of all finitesequences with positive rational coordinates; for a = (a1, . .

. , al) ∈Q(IN)+define the function fa on a sequence z1 < .

. .

< zl of normalized blocks byfa(z1, . .

. , zl) =lXi=1aizi.Let {a (k)} be an enumeration of Q(IN)+.

We will write fk for fa(k). Fixθk ↓0 satisfying θk < (1/2D)∥a (k)∥∞for k = 1, 2, .

. .. Let Z ∈B ∞(X) bethe stabilizing subspace for all the fks, constructed in 3.7.

Let [βk, δk] denotethe corresponding spectrum intervals for fk (k = 1, 2, . .

. ).The major role in our approach is played by two positive real valuedfunctions on Q(IN) defined via tilda-spectrum of the fks as follows.

Withfixed θk, Z, and [βk, δk], as above, we let, for a = a(k) ∈Q(IN)+,g(a) = βkandr(a) = δk. (4.1)These definitions can be naturally extended to all Q(IN), by setting, for ±a =(±a1, .

. .

, ±al), g(±a) = g(a) and r(±a) = r(a). We call the functions g andr the enveloping functions of X.4.2The main part of the proof of the theorem is contained in the followingproposition concerning the behaviour of functions g and r for spaces withbounded distortions.PropositionAssume that a Banach space X has bounded distortions andlet d(X) < D. With the notation from 4.1, either there exists 1 ≤p < ∞such that(1/D′)X|ai|p1/p ≤g(a) ≤r(a) ≤D′ X|ai|p1/pfor a ∈Q(IN),or(1/D′) max |ai| ≤g(a) ≤r(a) ≤D′ max |ai|for a ∈Q(IN).where D′ = 4D.21

4.3Assuming the truth of Proposition 4.2 let us complete the proof ofthe theorem.Proof of Therem 0.7 Let {xi} denote the block basis for the stabilizingsubspace Z, which has been fixed in 4.1.Assume that the conclusion ofProposition 4.2 is satisfied for some 1 ≤p ≤∞(with the obvious conventionfor p = ∞). We will then show that Z is asymptotic-lp (or asymptotic-c0, ifp = ∞).Fix n and fix ε = ε(n) > 0 to be defined later.

Let b (1), . .

. , b (M) be anε-net in the unit sphere S(lnp) of lnp in the l∞-norm, and assume without lossof generality that b (i) ∈Q(IN) for 1 ≤i ≤M.

Thus for every i there is k = kisuch that b (i) = a (k). Let N = max1≤i≤M ki.Let N < y1 < .

. .

< yn be arbitrary normalized blocks of {xk}. By 3.7and Proposition 4.2 we have, for every 1 ≤i ≤M,(1/D)∥b (i)∥p −θk≤g(b (i)) −θk ≤∥b (i)1 y1 + · · · + b (i)n yn∥≤r(b (i)) + θk ≤D∥b (i)∥p + θk.By the choice of θk this implies(1/2D) ≤∥b (i)1 y1 + · · · + b (i)n yn∥≤2DAn easy approximation argument shows that if ε is sufficiently small (itis enough to take ε = (4Dn)−1) then the latter estimates imply(1/4D) ≤∥c1y1 + · · · + cnyn∥≤4D,for any (c1, .

. .

, cn) ∈S(lnp). Thus y1 < .

. .

< yn are D′-equivalent to the unitvector basis in lnp, as required.✷5Inequalities for enveloping functionsThe proof of Proposition 4.2 is based on specific properties of functions gand r in spaces with bounded distortions, which will be established in thissection. In what follows we keep the notation from 4.1, and in particular,Z ∈B ∞(X) is the stabilizing subspace for the functions {fk}, constructedin 3.7.22

5.1A space Y with a basis {yi} is said to be asymptotically unconditionalif there exists a constant D′ such that for every n there exists N = N(n) suchthat for any normalized blocks N < z1 < . .

. < zn of {yi} and any sequenceof reals (c1, .

. .

, cn) we havesupεi=±1 ∥ε1c1z1 + . .

. + εncnzn∥≤D′ infεi=±1 ∥ε1c1z1 + .

. .

+ εncnzn∥. (5.1)LemmaLet X be a Banach space with bounded distortions.

Then it con-tains a subspace Y ∈B ∞(X) which is asymptotically unconditional.ProofAssume that X has a basis {xi}. Given α > 0, we will construct ablock basis {yi} for which (5.1) holds with the constant D′ = (1 + α) d(X)and N(n) = n.We will use a common and convenient notation that if I and J are inter-vals of positive integers then I < J means maxi∈I i < minj∈J j. Moreover,for x = Pi tixi ∈X, we set Ix = Pi∈I tixi.We may assume that X does not contain c0, otherwise the proof wouldbe finished.

For a positive integer n define the norm·n on X byxn = supnXi=1εiIix,where the supremum is taken over all intervals I1 < . .

. < In and all εi = ±1,i = 1, .

. .

, n.Clearly, ∥x∥≤xn ≤n∥x∥for x ∈X, so·n is anequivalent norm on X. We will show that the setAn =nx ∈S(X) |xn ≤(1 + α)o(5.2)is asymptotic in X.

Thus, by the Remark in 1.2 and (1.1), for every X′ ∈B ∞(X) there is Fn ∈B ∞(X′) such thatxn ≤(1 + α)D∥x∥for x ∈Fn.This leads to the inductive construction of subspaces X = F0 ⊃F1 ⊃. .

. ⊃Fn ⊃.

. .

with Fn ∈B ∞(Fn−1) and ∥x∥≤xn ≤(1 + α)D∥x∥for x ∈Fn(n = 1, 2, . .

. ).

It is easy to check that any block basis {yn} such that yn ∈S(Fn) satisfies (5.1) with D′ = (1 + α)D.To show that the set An given by (5.2) is asymptotic in X, let W =span [wi] ∈B ∞(X). Fix N > n to be defined later.

LetaN = supnNXj=1ηjwj | ηj = ±1o=NXj=1η0j wj.23

Since X does not contain c0, we have aN →∞, as N →∞.Set w = PNj=1 η0jwj. Given intervals I1 < .

. .

< In, let Li be the set of allj such that supp wj ⊂Ii and let Ki be the set of all j such that Iiwj ̸= 0, fori = 1, . .

. , n. For every εi = ±1, with i = 1, .

. .

, n, we havenXi=1εiIiw ≤nXi=1εi Xj∈Liη0jwj + 2n ≤aN + 2n.Thereforew/aNn ≤1 + 2n/aN.Thus, if aN > 2n/α, then w/aN ∈An ∩S(W) hence An is asymptotic in X.✷5.2Let X be a Banach space with bounded distortions, let Y ∈B ∞(X)be an asymptotically unconditional subspace of X, for some constant D′arbitrarily close to D which can be chosen later, and let Z ∈B ∞(Y ) be thestabilizing subspace for {fk} constructed in 3.7.The following lemma investigates the behaviour of enveloping functionsg and r : Q(IN) →IR .LemmaAssume that a Banach space X has bounded distortions and letd(X) < D. Theng(a) ≤r(a) ≤3Dg(a)for a ∈Q(IN).ProofThe left hand side inequality is obvious. To prove the right handside inequality, for i = 1, .

. .

, lk and k = 1, 2, . .

., let A (k)iand U (k)idenotethe ith asymptotic sets, constructed for the function fk and the subspace Z,as in 3.5.We will prove that, with fixed a = a (k) ∈Q(IN)+, we have∃G1 ∈B ∞(Z) ∀z1 ∈S(G1) ∃G2 ∈B ∞(G1, z1) ∀z2 ∈S(G2) . .

.fk(z1, z2, . .

. , zkl) ≤D(βk + θk).

(5.3)Applying (5.3) to vectors from the appropriate sets U (k)i, for i = 1, . .

. , lk,we getδk −θk ≤fk(z1, z2, .

. .

, zlk) ≤D(βk + θk).24

By the choice of θk from 4.1 we have θk ≤(1/2D)∥a (k)∥∞≤βk/2, thelatter inequality yieldsr(a) = δk ≤3Dβk ≤3Dg(a). (5.4)To prove that (5.3) holds, first note that since Z is asymptotically uncon-ditional, we can assume, without loss of generality that all the sets A (k)is aresymmetric about the origin.

Consider A (k)1⊂S(Z). By the Remark in 1.2there exists G1 ∈B ∞(Z) such that(1/D)(BX ∩G1) ⊂convA (k)1 .Thus for every z1 ∈S(G1) there exist w1, .

. .

, wm ∈A (k)1and t1, . .

. , tm,with tj > 0 and Pj tj = 1, such that (1/D)z1 = Pj tjwj.Now set G2,0 = G1 and proceed by induction in j = 1, .

. .

, m. For j ≥1consider the set A (k)2 (wj) ∩G2,j−1 constructed for the vector wj.Arguing asbefore we get a subspace G2,j ∈B ∞(G2,j−1, wj) such that(1/D)(BX ∩G2,j) ⊂conv(A (k)2 (wj)).Let G2 = G2,m ∈B ∞(G1, z1). For a fixed j = 1, .

. .

, m, we have, for anarbitrary vector z2 ∈S(G2),(1/D)z2 =m′Xn=1sn,jun,j,with un,j ∈A (k)2 (wj) and sn,j > 0 for n = 1, . .

. , m′ andPn sn,j = 1.We repeat the process lk times, for all subsets A (k)iwith i = 1, 2, .

. .

, lk.We then have, by the definitions of fk and of the sets A (k)i,fk(z1, z2, . .

. , zlk)=D ∥(1/D)a1z1 + (1/D)a2z2 + · · ·∥≤DXjtj ∥a1wj + (1/D)a2z2 + · · ·∥≤DXjtjXnsn,j ∥a1wj + a2un,j + · · ·∥≤.

. .

≤D(βk + θk),which is the required estimate (5.3).✷25

5.3Next lemma establishes general properties of the function r : Q(IN) →IR . It says that r can be extended in a natural way to the 1-unconditionaland 1-subsymmetric norm on the space IR (IN) of all real finite sequences.Moreover, denoting by {ei} the standard unit vector basis in IR (IN), theextended norm has certain blocking property.Recall that a norm | · | on IR (IN) is 1-subsymmetric if for every sequencea = (a1, a2, a3, .

. .) ∈IR (IN) we have |(a1, 0, .

. .

, 0, a2, 0, . .

. , 0, a3, .

. .

)| = |a|.LemmaThe function r(·) can be extended to the 1-unconditional and 1-subsymmetric norm on IR (IN). If {ui} is a block basis of the standard unitvector basis with r(ui) = 1 for i = 1, 2, .

. ., then for every a ∈IR (IN) we haveg(a) ≤r(d) ≤r(a).

(5.5)where for a = (a1, . .

. , al) = Pi aiei, by d we denote the sequence d = Pi aiui.Proof Consider r(·) as a function on Q(IN).

It is clearly positively homoge-neous and we will prove the triangle inequality by showing that if a, b ∈Q(IN)then r(a + b) ≤r(a) + r(b).Indeed, let a = a (i) = (a1, a2, . .

. ), b = a (j) = (b1, b2, .

. .

), and a+b = a (m).Fix an arbitrary η > 0. Pick arbitrary vectors wj ∈U (m)j, for j = 1, 2, .

. ..Thenr(a + b) −η ≤∥(a1 + b1)w1 + (a2 + b2)w2 + · · ·∥,moreover,∥a1w1 + a2w2 + · · · ∥≤r(a) + ηand∥b1w1 + b2w2 + · · ·∥≤r(b) + η.Since η > 0 is arbitrary, this shows the triangle inequality.Recall that for a sequence a = (a1, a2, .

. .

), we set ±a = (±a1, ±a2, . .

. ).Then we have r(a) = r(±a), hence the norm r(·) is 1-unconditional.

It isalso clearly 1-subsymmetric.To prove (5.5), let ui =Pki+1j=ki+1 bjej, for some 0 ≤k1 < k2 < . .

., be ablock basis with rational coefficients of the standard unit vector basis, withr(ui) = 1 for i = 1, 2, . .

.. Let a = (a1, . .

. , al) =Pi aiei, then d =Pi aiui =(d1, .

. .

, dkl+1), where dj = aibj for ki < j ≤ki+1 and i = 1, 2, . .

..26

Let η > 0. There exist vectors w1 < w2 < .

. .

< wkl+1 in appropriateasymptotic sets such thatr(d) −η ≤kl+1Xj=1djwj.Since all the vectors belong to Z, we also have, from the form of d,ci =ki+1Xj=ki+1djwj ≤|ai|(r(ui) + η) = |ai|(1 + η),for i = 1, . .

. , l. By the triangle inequality and the unconditionality of thenorm r(·) this implies r(c1, .

. .

, cl) ≤r(a)(1 + η).Setting wi = (1/ci) Pki+1j=ki+1 djwj we getr(d) −η ≤lXi=1ciwi ≤r(c1, . .

. , cl) + ηwhich combined with the previous inequality shows the right hand side of(5.5).The proof of the left hand side inequality is similar.

For an arbitraryη > 0, pick vectors w′1 < w′2 < . .

. < w′kl+1 in appropriate asymptotic sets(for appropriate functions fi) such that for every i = 1, .

. .

, l one has1 −η = r(ui) −η ≤ki+1Xj=ki+1bjw′j ≤r(ui) + η = 1 + η.ThenXiaiki+1Xj=ki+1bjw′j =kl+1Xj=1djw′j ≤r(d) + η.On the other hand, setting vi = Pki+1j=ki+1 bjw′j for i = 1, . .

. , l we have1 −η ≤∥vi∥≤1 + η. ThenXiaiki+1Xj=ki+1bjw′j =lXi=1aivi ≥(g(a) −η)(1 −η).Combining the last two estimates we complete the proof of the left hand sideof (5.5).✷27

5.4Now we can easily complete the proof of Proposition 4.2.Proof of Proposition 4.2Let L denotes the completion of (IR (IN), r(·)),and let {ei} be the standard unit vector basis in L. Lemmas 5.2 and 5.3yield that all normalized block bases of {ei} are (3D)-equivalent to {ei}. ByZippin’s theorem, this implies that the {ei} is equivalent to the standard unitvector basis in lp, for some 1 ≤p < ∞or in c0.

Moreover, the equivalenceconstant depends on D only.✷6Asymptotic lp spaces, general propertiesWe conclude this paper with few simple remarks on general asymptotic lpspaces. To avoid tiresome repetitions, when talking about spaces lp or asymp-totic lp, respectively, we adopt the convention that the case of p = ∞corre-sponds to the space c0 or asymptotic c0, respectively.Let Y with a basis {yi} be an asymptotic lp space for 1 ≤p ≤∞, andlet λp(Y ) be the asymptotic lp constant, as defined in 0.7.6.1It is well-known and easy to see that any block subspace of lp is com-plemented.

The same is true in Tsirelson space and in its convexifications (cf.e.g., [C-S]), although in this case the argument is much more complicated. Ananalogous fact for arbitrary asymptotic lp space says that finite-dimensionalblock subspaces far out are uniformly complemented.More precisely, for C > λp(Y ), if N(n) = N < z1 < .

. .

< zn arenormalized blocks C-equivalent to the standard unit vector basis in lnp, thenthere exists a projection P from Y onto span [zi]ni=1 with ∥P∥≤2C2.Indeed, pick z∗i ∈Y ∗such that ∥z∗i ∥= z∗i (zi) = 1 for i = 1, . .

. , n. LetN < E1 < .

. .

< En be intervals of positive integers such that supp zi ⊂Ei,for i = 1, . .

. , n, and that the union of all the Eis is an interval.

For x ∈YsetPx =nXi=1z∗i (Eix)zi.Since Ejzi = 0 if i ̸= j, then P is a projection. Moreover, we have∥Px∥≤C nXi=1|z∗i (Eix)|p1/p ≤C nXi=1∥Eix∥p1/p28

≤C2nXi=1∥Eix∥( Eix∥Eix∥) ≤C2nXi=1Eix ≤2C2∥x∥,as required.6.2If Y is an asymptotic lp space (for 1 < p ≤∞) then the dual Y ∗is anasymptotic lp′ space. This follows from 6.1 by a general duality argument.A direct calculation is just as standard and simple and we leave it to thereader.6.3It is easy to observe that if p > 1, the basis in Y is shrinking andif p < ∞, the basis is boundedly complete.Hence for 1 < p < ∞, anasymptotic lp space is reflexive.Assume to the contrary that the basis {yi} is not shrinking.

There existsx∗∈Y ∗with ∥x∗∥= 1, and δ > 0, and a normalized block basis {ui} of {yi}such that |x∗(ui)| > δ. Fix n to be defined later.

Then for every k we havek+n−1Xi=kui ≥x∗k+n−1Xi=kui ≥nδ.On the other hand, if k is large enough, the left hand side is smaller than orequal to C n1/p. Chosing appropriate n we get a contradition, if p > 1.Assume the basis is not boundedly complete.

Then there exists normal-ized block basis {ui} of {yi} such that supn ∥Pni=1 ui∥= M < ∞. On theother hand if uk < .

. .

< uk+n−1 is far enough, thenk+n−1Xi=kui ≥(1/C) n1/p.If p < ∞, we again come to a contradition by an appropriate choice of n.6.4The notion of asymptotic lp spaces is fundamentally an isomorphicconcept and it cannot be reduced to a (1 + ε)- isometric one. As mentionedin 0.7, to the contrary to the local concept of finite representability of lp,asymptotic lp does not imply any related almost isometric property of ablock subspace.

To be more precise let us discuss the quantity λp(Y ) in moredetail.29

Note that the isometric condition λp(Y ) = 1 is equivalent to the factthat for every ε > 0 and for every n there exists N = N(ε, n) such thatany normalized blocks N < z1 < z2 < . .

. < zn are (1 + ε)-equivalent to thestandard unit vector basis in lnp.

(In 0.7 we called such a space an almostisometric asymptotic lp space. )Recall that if a space Y merely satisfies a weaker condition: for every ε > 0there exists N = N(ε) such that any two normalized blocks N < z1 < z2are (1 + ε)-equivalent to the standard unit vector basis in l2p, then, for everyε > 0, Y contains an (1 + ε)-isomorphic copy of the lp-space.

Let us sketchthis well-known and standard argument.Given ε > 0, fix εi ↓0 such thatQi(1 + εi) ≤(1 + ε).By an easyinduction pick a sequence of normalized blocks u1 < u2 < . .

. < ui < .

. .such that N(εi) < ui for i = 1, 2, .

. .. Then the block basis {ui} is (1 + ε)-equivalent to the standard unit vector basis in lp.Indeed, for any finitesequence of scalars {ai} we have∞Xi=1aiui≤(1 + ε1) |a1|p +∞Xi=2aiuip!1/p≤(1 + ε1)(1 + ε2) |a1|p + |a2|p +∞Xi=3aiuip!1/p≤.

. .

≤Yi(1 + εi) ∞Xi=1|ai|p1/p.In a similar way we get the lower estimate, hence(1 + ε)−1 ∞Xi=1|ai|p1/p ≤∞Xi=1aiui ≤(1 + ε) ∞Xi=1|ai|p1/p,as required.Clearly, λp(Y ) is an isomorphic invariant. However, there exist spaces Ysuch that λp(Y ) < ∞but there is no equivalent norm·on Y such that forsome block subspace Z ∈B ∞(Y ) the equality λp(Y,·) = 1 would hold.The construction above obviously yields that this is true for every asymptoticlp space which does not contain subspaces isomorphic to lp.

In particular,Tsirelson space and its convexifications have this property.30

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